Question

Find the smallest number which should be added to the smallest number divisible by 6, 9 and 15 to make it a perfect square.

• A. 10
• B. 9
• C. 19
• D. 21

Hint:

Always calculate the LCM first. Once you have the base number, look up the nearest perfect square that is greater than the base number and take the difference.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem requires finding the smallest positive integer that must be added to a base number.
The base number is the smallest positive integer that is completely divisible by 6, 9, and 15, which is their Least Common Multiple (LCM).
After finding this LCM, we identify the next immediate perfect square and calculate the difference to find the number to be added.

Step 2: Key Formula or Approach:
We use the prime factorization method to find the LCM of the given numbers:
\[ 6 = 2 \times 3 \]
\[ 9 = 3^2 \]
\[ 15 = 3 \times 5 \]
The LCM is the product of the highest power of each prime factor involved:
\[ \text{LCM} = 2^1 \times 3^2 \times 5^1 \]

Step 3: Detailed Explanation:
1. First, we compute the Least Common Multiple (LCM) of 6, 9, and 15 to find the smallest number divisible by all three:
\[ \text{LCM} = 2 \times 9 \times 5 = 90 \] So, the smallest number divisible by 6, 9, and 15 is 90.
2. Now, we need to find the smallest number that should be added to 90 to make it a perfect square.
3. Let us list the perfect squares that lie close to 90:
\[ 9^2 = 81 \]
\[ 10^2 = 100 \]
\[ 11^2 = 121 \]
4. Since we must add a positive number to 90, the target perfect square must be greater than 90.
5. The smallest perfect square greater than 90 is 100.
6. Let the number to be added be \( x \).
7. We set up the relation: \( 90 + x = 100 \).
8. Solving for \( x \):
\[ x = 100 - 90 = 10 \] 9. Note that the circled option in the question image is 9, which is incorrect because \( 90 + 9 = 99 \), which is not a perfect square. The correct value is 10, which corresponds to option (A).

Step 4: Final Answer:
The smallest number that should be added is 10.